Integrand size = 22, antiderivative size = 76 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x} \, dx=a A \sqrt {a+b x^2}+\frac {1}{3} A \left (a+b x^2\right )^{3/2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b}-a^{3/2} A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {457, 81, 52, 65, 214} \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x} \, dx=-a^{3/2} A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {1}{3} A \left (a+b x^2\right )^{3/2}+a A \sqrt {a+b x^2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b} \]
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Rule 52
Rule 65
Rule 81
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{3/2} (A+B x)}{x} \, dx,x,x^2\right ) \\ & = \frac {B \left (a+b x^2\right )^{5/2}}{5 b}+\frac {1}{2} A \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{3} A \left (a+b x^2\right )^{3/2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b}+\frac {1}{2} (a A) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right ) \\ & = a A \sqrt {a+b x^2}+\frac {1}{3} A \left (a+b x^2\right )^{3/2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b}+\frac {1}{2} \left (a^2 A\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right ) \\ & = a A \sqrt {a+b x^2}+\frac {1}{3} A \left (a+b x^2\right )^{3/2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b}+\frac {\left (a^2 A\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{b} \\ & = a A \sqrt {a+b x^2}+\frac {1}{3} A \left (a+b x^2\right )^{3/2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b}-a^{3/2} A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x} \, dx=\frac {\sqrt {a+b x^2} \left (20 a A b+3 a^2 B+5 A b^2 x^2+6 a b B x^2+3 b^2 B x^4\right )}{15 b}-a^{3/2} A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
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Time = 2.80 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {B \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}+A \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\) | \(71\) |
pseudoelliptic | \(\frac {-3 a^{\frac {3}{2}} b A \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )+4 \left (\frac {x^{2} \left (\frac {3 x^{2} B}{5}+A \right ) b^{2}}{4}+a \left (\frac {3 x^{2} B}{10}+A \right ) b +\frac {3 a^{2} B}{20}\right ) \sqrt {b \,x^{2}+a}}{3 b}\) | \(73\) |
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Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x} \, dx=\left [\frac {15 \, A a^{\frac {3}{2}} b \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, B b^{2} x^{4} + 3 \, B a^{2} + 20 \, A a b + {\left (6 \, B a b + 5 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{30 \, b}, \frac {15 \, A \sqrt {-a} a b \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, B b^{2} x^{4} + 3 \, B a^{2} + 20 \, A a b + {\left (6 \, B a b + 5 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, b}\right ] \]
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Time = 10.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x} \, dx=\frac {\begin {cases} \frac {2 A a^{2} \operatorname {atan}{\left (\frac {\sqrt {a + b x^{2}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 A a \sqrt {a + b x^{2}} + \frac {2 A \left (a + b x^{2}\right )^{\frac {3}{2}}}{3} + \frac {2 B \left (a + b x^{2}\right )^{\frac {5}{2}}}{5 b} & \text {for}\: b \neq 0 \\A a^{\frac {3}{2}} \log {\left (B a^{\frac {3}{2}} x^{2} \right )} + B a^{\frac {3}{2}} x^{2} & \text {otherwise} \end {cases}}{2} \]
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Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x} \, dx=-A a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A + \sqrt {b x^{2} + a} A a + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{5 \, b} \]
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Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x} \, dx=\frac {A a^{2} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{4} + 5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{5} + 15 \, \sqrt {b x^{2} + a} A a b^{5}}{15 \, b^{5}} \]
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Time = 5.50 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x} \, dx=\frac {A\,{\left (b\,x^2+a\right )}^{3/2}}{3}+\frac {B\,{\left (b\,x^2+a\right )}^{5/2}}{5\,b}-A\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )+A\,a\,\sqrt {b\,x^2+a} \]
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